In Simachew's "A Survey on Euclidean Number Fields", he said that Gauss used the existence of a Euclidean algorithm in Gaussian integers to prove that it has unique factorization. Also, he said that Wantzel observed that norm-Euclideanity implies unique factorization in cyclotomic integers. And Kummer used this to prove some stuff about some cyclotomic integers, which then paved the way to the desire of determining Euclideanity of number fields. In Lemmermeyer's survey, "The Euclidean Algorithm in Algebraic Number Fields", he said that Dirichlet was "the first mathematician who emphasized that the existence of a Euclidean algorithm implied unique factorization" in 1847. However, Wantzel's observation was also around 1847.

So my question is, in this case, was it Dirichlet who emphasized that it is true "in general"? Like not just norm-Euclideanity, and not just on a specific ring of integers?

  • 1
    $\begingroup$ Most probably it was Dirichlet. The general notion of an Euclidean domain did not exist before then, But in fact Gauss did all the job. $\endgroup$
    – markvs
    May 22 '21 at 19:24
  • $\begingroup$ @MarkSapir Do you know a reference for this? Thank you very much for answering :) $\endgroup$
    – Butterfly
    May 23 '21 at 20:14
  • $\begingroup$ It's because in Lemmermeyer's paper, he just stated it in the introduction but didn't put a reference, sadly. $\endgroup$
    – Butterfly
    May 23 '21 at 20:16

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